×

Nonvanishing of Siegel Poincaré series. (English) Zbl 1300.11048

Summary: We prove that, under suitable conditions, certain Siegel Poincaré series of exponential type of even integer weight and degree 2 do not vanish identically. We also find estimates for twisted Kloosterman sums and Salié sums in all generality.

MSC:

11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
11L05 Gauss and Kloosterman sums; generalizations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Böcherer S., Kohnen W.: Estimates for Fourier coefficients of Siegel cusp forms. Math. Ann. 297, 499–517 (1993) · Zbl 0787.11017 · doi:10.1007/BF01459512
[2] Christian U.: Über Hilbert-Siegelsche Modulformen und Poincarésche Reihen. Math. Ann. 148, 257–307 (1962) · Zbl 0141.27601 · doi:10.1007/BF01451138
[3] Das S.: Nonvanishing of Jacobi Poincaré series. J. Aust. Math. Soc. 89, 165–179 (2010) · Zbl 1248.11035 · doi:10.1017/S1446788710001576
[4] Das, S.: Nonvanishing of Jacobi Poincaré series. http://arxiv.org/abs/0910.4303v2
[5] Das, S., Kohnen, W., Sengupta, J.: Nonvanishing of Siegel Poincaré series II (preprint, 2011)
[6] Eichler M., Zagier D.: The Theory of Jacobi Forms. Progress in Mathematics, vol. 55. Boston (1985) · Zbl 0554.10018
[7] Ireland K., Rosen M.: A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics, vol. 84. Springer, Berlin (1990) · Zbl 0712.11001
[8] Iwaniec H.: Fourier coefficients of modular forms of half-integral weight. Invent. Math. 87, 385–401 (1987) · Zbl 0606.10017 · doi:10.1007/BF01389423
[9] Iwaniec, H., Kowalski, E.: Analytic Number Theory, vol. 53. A.M.S. Colloquium Publications, Providence (2004) · Zbl 1059.11001
[10] Kitaoka Y.: Fourier coefficients of Siegel cusp forms of degree two. Nagoya. Math. J. 93, 149–171 (1984) · Zbl 0531.10031
[11] Klingen H.: Introductory Lectures on Siegel Modular Forms. Cambridge Studies in Advanced Mathematics, vol. 20. Cambridge University Press, Cambridge (1990) · Zbl 0693.10023
[12] Kohnen W.: Fourier coefficients of modular forms of half integral weight. Math. Ann. 271, 237–268 (1985) · Zbl 0553.10020 · doi:10.1007/BF01455989
[13] Kohnen W.: On Poincaré series of exponential type on Sp2. Abh. Math. Sem. Hamburg 63, 283–297 (1993) · Zbl 0790.11038 · doi:10.1007/BF02941348
[14] Kohnen W.: Lifting modular forms of half-integral weight to Siegel modular forms of even genus. Math. Ann. 322, 787–809 (2002) · Zbl 1004.11020 · doi:10.1007/s002080100285
[15] Krieg A.: Modular Forms on Half-Spaces of Quaternions. Lecture Notes in Mathematics, vol. 1143. Springer, Berlin (1985) · Zbl 0564.10032
[16] Mozzochi C.J.: On the nonvanishing of Poincaré series. Proc. Edinburgh Math. Soc. 32, 133–137 (1989) · Zbl 0649.10018 · doi:10.1017/S0013091500006982
[17] Muić, G.: On the nonvanishing of certain Modular forms. Int. J. Number Theory (to appear) · Zbl 1267.11039
[18] Rankin R.A.: The vanishing of Poincaré series. Proc. Edinburgh Math. Soc. 23, 151–161 (1980) · Zbl 0454.10013 · doi:10.1017/S0013091500003035
[19] Salié H.: Über die Kloostermanschen Summen S(u, v; q). Math. Z. 34, 91–101 (1931) · JFM 57.0211.01 · doi:10.1007/BF01180579
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.